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      "cell_type": "code",
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      "source": [
        "%matplotlib inline"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "\n# Kernel PCA\n\n\nThis example shows that Kernel PCA is able to find a projection of the data\nthat makes data linearly separable.\n\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
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      "source": [
        "print(__doc__)\n\n# Authors: Mathieu Blondel\n#          Andreas Mueller\n# License: BSD 3 clause\n\nimport numpy as np\nimport matplotlib.pyplot as plt\n\nfrom sklearn.decomposition import PCA, KernelPCA\nfrom sklearn.datasets import make_circles\n\nnp.random.seed(0)\n\nX, y = make_circles(n_samples=400, factor=.3, noise=.05)\n\nkpca = KernelPCA(kernel=\"rbf\", fit_inverse_transform=True, gamma=10)\nX_kpca = kpca.fit_transform(X)\nX_back = kpca.inverse_transform(X_kpca)\npca = PCA()\nX_pca = pca.fit_transform(X)\n\n# Plot results\n\nplt.figure()\nplt.subplot(2, 2, 1, aspect='equal')\nplt.title(\"Original space\")\nreds = y == 0\nblues = y == 1\n\nplt.scatter(X[reds, 0], X[reds, 1], c=\"red\",\n            s=20, edgecolor='k')\nplt.scatter(X[blues, 0], X[blues, 1], c=\"blue\",\n            s=20, edgecolor='k')\nplt.xlabel(\"$x_1$\")\nplt.ylabel(\"$x_2$\")\n\nX1, X2 = np.meshgrid(np.linspace(-1.5, 1.5, 50), np.linspace(-1.5, 1.5, 50))\nX_grid = np.array([np.ravel(X1), np.ravel(X2)]).T\n# projection on the first principal component (in the phi space)\nZ_grid = kpca.transform(X_grid)[:, 0].reshape(X1.shape)\nplt.contour(X1, X2, Z_grid, colors='grey', linewidths=1, origin='lower')\n\nplt.subplot(2, 2, 2, aspect='equal')\nplt.scatter(X_pca[reds, 0], X_pca[reds, 1], c=\"red\",\n            s=20, edgecolor='k')\nplt.scatter(X_pca[blues, 0], X_pca[blues, 1], c=\"blue\",\n            s=20, edgecolor='k')\nplt.title(\"Projection by PCA\")\nplt.xlabel(\"1st principal component\")\nplt.ylabel(\"2nd component\")\n\nplt.subplot(2, 2, 3, aspect='equal')\nplt.scatter(X_kpca[reds, 0], X_kpca[reds, 1], c=\"red\",\n            s=20, edgecolor='k')\nplt.scatter(X_kpca[blues, 0], X_kpca[blues, 1], c=\"blue\",\n            s=20, edgecolor='k')\nplt.title(\"Projection by KPCA\")\nplt.xlabel(r\"1st principal component in space induced by $\\phi$\")\nplt.ylabel(\"2nd component\")\n\nplt.subplot(2, 2, 4, aspect='equal')\nplt.scatter(X_back[reds, 0], X_back[reds, 1], c=\"red\",\n            s=20, edgecolor='k')\nplt.scatter(X_back[blues, 0], X_back[blues, 1], c=\"blue\",\n            s=20, edgecolor='k')\nplt.title(\"Original space after inverse transform\")\nplt.xlabel(\"$x_1$\")\nplt.ylabel(\"$x_2$\")\n\nplt.tight_layout()\nplt.show()"
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